Fourier Coefficients and Automorphic Discrete Spectrum of Classical Groups. Dihua Jiang University of Minnesota

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1 Fourier Coefficients and Automorphic Discrete Spectrum of Classical Groups Dihua Jiang University of Minnesota KIAS, Seoul November 16, 2015

2 Square-Integrable Automorphic Forms G a reductive algebraic group defined over a number field F. A is the ring of adeles of F. X G := G(F )\G(A) 1, where G(A) 1 := χ X (G) ker χ A. L 2 (X G ) denotes the space of functions: φ : X G C such that G(Q)\G(A) 1 φ(g) 2 dg <. A 2 (G) is the set of equivalence classes of irreducible unitary representations of G(A) occurring in the discrete spectrum L 2 disc (X G). A cusp (G) is the subset of A 2 (G) consisting of those automorphic representations of G(A) occurring in the cuspidal spectrum L 2 cusp(x G ).

3 Theory of Endoscopic Classification Theorem Let G be an F -quasisplit classical group and G be a pure inner form of G over F. For any π A cusp (G), there is a global Arthur parameter ψ Ψ 2 (G ), which is G-relevant, such that π Π ψ (G) where Π ψ (G) is the global Arthur packet of G associated to ψ. G is a quasisplit SO or a split Sp, Arthur s book 2013, including Chapter 9 for non-quasisplit case. G is a quasiplit U m, C.-P. Mok Mem. of AMS, G is GSp and GSO, Bin Xu (2014, 2015) (student of Arthur). G is a pure inner form of quasisplit U m, preprint of 2014 by Kaletha-Minguez-Shin-White. For more general unitary groups, it is their work in progress.

4 Global Arthur Parameters Ψ 2 (G): Examples G = SO 2n+1, F -split, and (G ) = Sp 2n (C). Each ψ Ψ 2 (G ) (global Arthur parameters) is written as a formal sum of simple Arthur parameters: ψ = ψ 1 ψ 2 ψ r where ψ i = (τ i, b i ), with τ i A cusp (GL ai ); a i, b i 1; and r i=1 a ib i = 2n. If i j, either τ i = τj or b i b j, with the parity condition that a i b i is even and ψ i Ψ 2 (SO a i b i +1 ). Endoscopy Structure: 2n = r i=1 a i b i, SO a 1 b 1 +1 SO a r b r+1 = SO 2n+1 Π ψ1 ( ) Πψr ( ) = Π ψ ( )

5 Global Arthur Parameters Ψ 2 (G): Examples G = Sp 2n, F -split, and (G ) = SO 2n+1 (C). Each ψ Ψ 2 (G ) is written as a formal sum of simple Arthur parameters: ψ = ψ 1 ψ 2 ψ r where ψ i = (τ i, b i ), with τ i A cusp (GL ai ); a i, b i 1; r i=1 a ib i = 2n + 1; and r i=1 ωb i τ i = 1. If i j, either τ i = τj or b i b j, with the parity: 1 If a i b i is even, then ψ i Ψ 2 (SO a ib i ); 2 If a i b i is is odd, then ψ i Ψ 2 (Sp a ib i 1). Endoscopy Structure: 2n + 1 = r i=1 a i b i, a i b i =2l i SO 2l i a j b j =2l j +1 Sp 2l j = Sp 2n ai b i =2l i Πψi ( ) aj b j =2l j +1 Π ψj ( ) = Π ψ ( )

6 Global Arthur Parameters Ψ 2 (G): Examples A parameter ψ = ψ 1 ψ 2 ψ r Ψ 2 (G ) is generic if b 1 = = b r = 1. Generic global Arthur parameters φ Φ 2 (G ) are: φ = (τ 1, 1) (τ 2, 1) (τ r, 1) with τ i A cusp (GL ai ) that τ i = τj if i j. They are of either symplectic or orthogonal type, depending on G. The pure inner forms of G = SO m are G = SO m (V, q) for non-deg. quad. spaces (V, q) over F with the same dimension and discriminant. If G is a pure inner form of G, then L G = L G. For φ Φ 2 (G ), the endoscopic classification may define the global Arthur packet Π φ (G ) and also define the global Arthur packet Π φ (G), which is non-empty if φ is G-relevant.

7 Endoscopic Classification and Langlands Functoriality Ψ 2 (G ) G ψ (1) A 2 (G) Π ψ (G) Πψ (G ) A 2 (G ) Remarks: In W. Schmid s 70th birthday conference, Arthur explains that the stable trace formula method obtains the character relations among those objects. Arthur-Schmid Question: How to construct modules for π in A 2 (G) Π ψ (G) in general?

8 Problems Based on the Endoscopic Classification By using the automorphic descent of Ginzburg-Rallis-Soudry, one can show that Π φ (G ) A cusp (G ). Problem (1): What happens if G is replaced by general G? I addressed this problem in my lecture at the Roger Howe 70th birthday conference, which covers mainly my joint work with Lei Zhang, and answers a Question of Arthur-Schmid on explicit constructions of automorphic modules for cuspidal automorphic representations in terms of their global Arthur parameters. The goal is to construct all members in Π φ (G) A cusp (G). Problem (2): What happens if the generic Arthur parameters φ are replaced by general Arthur parameters ψ? This is part of my joint project with Baiying Liu, and will be the main focus of my lecture today.

9 Problems Based on the Endoscopic Classification First of all, for classical groups G and generic Arthur parameters φ in Φ 2 (G ), which is G-relevant, one can show that Π φ (G) A 2 (G) is contained in A cusp (G). However, for a general Arthur parameter ψ, the set Π ψ (G) A cusp (G) may be empty, even if the set Π ψ (G) A 2 (G) is not empty. Hence we are interested in the conditions on ψ that the set Π ψ (G) A cusp (G) is empty; and if it is not empty, we want to construct members in the set Π ψ (G) A cusp (G). The idea is to use Fourier coefficients of automorphic representations to determine their automorphic wave-front set. This leads to its connections with the notion of singular automorphic forms in terms of the representation-theoretic formulation of Roger Howe and the work of Jianshu Li.

10 Fourier Coefficients and Nilpotent Adjoint Orbits G is an F -quasi-split classical group and g is the Lie algebra. Let N G be dimension for the defining embedding G GL(N G ). Over algebraic closure F of F, all the nilpotent elements in g (F ) form a conic algebraic variety, called the nilcone N (g ). Under the adjoint action of G, N (g ) decomposes into finitely many adjoint G -orbits O, which are parameterized by the corresponding partitions of N = N G of type G. Over F, each F -orbit reduces to an F -stable adjoint G (F )-orbits O st, and hence the F -stable adjoint orbits in the nilcone N (g ) are also parameterized by the corresponding partitions of an integer N = N G of type G.

11 Fourier Coefficients and Nilpotent Adjoint Orbits For X N (g ), use sl 2 -triple (over F ) to define a unipotent subgroup V X and a character ψ X. Let {X, H, Y } be an sl 2 -triple (over F ). Under the adjoint action of ad(h), g = g r g 2 g 1 g 0 g 1 g 2 g r. Ad(G )(Y ) g 2 and Ad(G )(X) g 2 are Zariski dense in g 2 and g 2, respectively. Take V X to be the unipotent subgroup of G such that the Lie algebra of V X is equal to i 2 g i. Let ψ F be a non-trivial additive character of F \A. The character ψ X of V X (F ) or V X (A) is defined by ψ X (v) = ψ F (tr(y log(v))).

12 Fourier Coefficients and Nilpotent Adjoint Orbits The Fourier coefficient of ϕ π A 2 (G ) is defined by F ψ X (ϕ)(g) := ϕ(vg)ψ X (v) 1 dv. V X (F )\V X (A) Since ϕ is automorphic, the nonvanishing of F ψ X (ϕ) depends only on the G (F )-adjoint orbit O X of X. The set n(ϕ) := {X N (g) F ψ X (ϕ) 0} is stable under the G (F )-adjoint action. Denoted by p(ϕ) the set of partitions p of N G of type G corresponding to the F -stable orbits Op st that have non-empty intersection with n(ϕ). p m (ϕ) is the set of all maximal partitions in p(ϕ), according to the partial ordering of partitions.

13 Maximal Fourier Coefficients of Automorphic Forms For π A 2 (G), denote by p m (π) the set of maximal members among p m (ϕ) for all ϕ π. One may think of the set p m (π) as an algebraic version of the wave-front set of π in the sense of Roger Howe. Basic Problems: How to determine the set p m (π) in terms of other invariants of π? What can one say about π if one knows the structure of p m (π)? Folklore Conjecture: For any irreducible automorphic representation π of G, the set p m (π) contains only one partition.

14 Maximal Fourier Coefficients of Automorphic Forms Examples: G = GL n, the G(F )-stable orbits in N (g) are parameterized by partitions of n. Theorem (Piatetski-Shapiro; Shalika): If π A 2 (GL n ) is cuspidal, p m (π) = {[n]}. This says that any irreducible cuspidal automorphic representation has a nonzero Whittaker-Fourier coefficient. What happens if π A 2 (GL n ) is not cuspidal? Moeglin-Waldspurger Theorem: Any π A disc (GL n ) has form (τ, b) (Speh residue with cuspidal support τ b ), where τ A cusp (GL a ) and n = ab. If π = (τ, b), then p m (π) = {[a b ]} (Ginzburg(2006), J.-Liu(2013) gives a complete global proof). In particular, the Folklore Conjecture is verified for all π A 2 (GL n )!

15 Maximal Fourier Coefficients and Arthur Parameters How to understand this in terms of Arthur parametrization? (τ, b) has the Arthur parameter ψ = (τ, b). The partition attached to ψ is p ψ := [b a ] and p m ( (τ, b)) = {[a b ]}. η([b a ]) = [a b ] is given by the Barbasch-Vogan duality η from GL n to GL n. Take an Arthur parameter for GL n : for τ i A cusp (GL ai ), ψ = (τ 1, b 1 ) (τ 2, b 2 ) (τ r, b r ). The partition attached to ψ is p ψ = [b a 1 1 ba 2 2 bar r ]. The Arthur representation is an isobaric sum π ψ = (τ 1, b 1 ) (τ 2, b 2 ) (τ r, b r ). Conjecture: p m (π ψ ) = {η gl n,gl n (p ψ )}.

16 Maximal Fourier Coefficients and Arthur Parameters For ψ = ψ 1 ψ 2 ψ r Ψ 2 (G ), where ψ i = (τ i, b i ) with τ i A cusp (GL ai ) and b i 1, p ψ = [b a 1 1 bar r ] is the partition of N G attached to (ψ, (G ) ) and η(p ψ ) is the Barbasch-Vogan duality of p ψ from (G ) to G. Conjecture (J.-2014): (1) For every π Π ψ (G ) A 2 (G ), any partition p p m (π) has the property that p η(p ψ ). (2) There exists at least one member π Π ψ (G ) A 2 (G ) having the property that η(p ψ ) p m (π). Remark: For a pure inner form G of G, assume that the global Arthur parameter ψ is G-relevant and the Barbasch-Vogan duality η(p ψ ) is a G-relevant partition of N G = N G of type G. Then the above Conjecture should also hold for G and G-relevant Arthur parameters.

17 Examples of the Barbasch-Vogan duality G = SO 2n+1 and 2n = ab; Take ψ = (τ, b) for τ A cusp (GL a ), and { 2l, if τ is orthogonal, b = 2l + 1, if τ is symplectic. p ψ = [b a ] is the partition of 2n of type (ψ, Sp 2n (C)). The Barbasch-Vogan duality is given as follows: [(a + 1)a b 2 (a 1)1] if b = 2l and a is even; η(p ψ ) = [a b 1] if b = 2l and a is odd; [(a + 1)a b 1 ] if b = 2l + 1.

18 Examples of the Barbasch-Vogan duality Take G = Sp 2n and ψ = (τ, 2b + 1) r i=2 (τ i, 1) Ψ 2 (G). p ψ = [(2b + 1) a (1) 2m+1 a ] with 2m + 1 = (2n + 1) 2ab. When a 2m and a is even, η(p ψ ) =η([(2b + 1) a (1) 2m+1 a ]) = [(2b + 1) a (1) 2m a ] t =[(a) 2b+1 ] + [(2m a)] = [(2m)(a) 2b ]. When a 2m and a is odd, η(p ψ ) =η([(2b + 1) a (1) 2m+1 a ]) =([(2b + 1) a (1) 2m a ] Sp2n ) t =[(2b + 1) a 1 (2b)(2)(1) 2m a 1 ] t =[(a 1) 2b+1 ] + [(1) 2b ] + [(1) 2 ] + [(2m 1 a)] =[(2m)(a + 1)(a) 2b 2 (a 1)].

19 Remarks on the Conjecture It is true when G = GL n and ψ is an Arthur parameter for the discrete spectrum. If φ Φ 2 (G) is generic, i.e. b 1 = = b r = 1, the partition p φ = [1 N G]. The Barbasch-Vogan duality of p φ is η([1 N G]) = [N G ] G. It is clear that the partition η([1 N G]) is G-relevant if G = G is quasi-split. The conjecture claims that any generic global Arthur packet contains a generic member for quasi-split G, and hence implies the global Shahidi conjecture on genericity of tempered packets. This special case can be proved by the Arthur-Langlands transfer from G to GL NG and the Ginzburg-Rallis-Soudry descent.

20 Remarks on the Conjecture The conjecture has been verified for various cases of Sp 2n through my joint work with Liu. The conjecture provides an upper bound partition for π Π ψ (G) A 2 (G), with a given Arthur parameter ψ. It is our work in progress to obtain a lower bound partition for π Π ψ (G) A 2 (G), with a given Arthur parameter ψ. It is very interesting, but harder problem to determine p m (π) for general automorphic members π Π ψ (G) A 2 (G) for each given Arthur parameter ψ! The theory of singular automorphic forms of Howe and Li provides a lower bound partition for π A cusp (G). It is not hard to find that the lower bound partition provided by the theory of Howe and Li may not be the best for all π A cusp (G).

21 Singular Partitions and Rank in the Sense of Howe G = Sp 2n. Take P n = M n U n to be the Siegel parabolic of Sp 2n, with M n = GLn and the elements of U n are of form ( ) In X u(x) =. 0 I n By the Pontryagin duality, one has U n (F )\U n (A) = Sym 2 (F n ). For a fixed nontrivial additive character ψ F of F \A, a T Sym 2 (F n ) corresponds to the character ψ T by ψ T (u(x)) := ψ F (tr(t wx)), with w anti-diagonal, and the action of GL n on Sym 2 (F n ) is induced from its adjoint action on U n.

22 Singular Partitions and Rank in the Sense of Howe For an automorphic form ϕ on Sp 2n (A), the T - or ψ T -Fourier coefficient is defined by F ψ T (ϕ)(g) := ϕ(u(x)g)ψ 1 T (u(x))du(x). U m(f )\U n(a) An automorphic form ϕ on Sp 2n (A) is called non-singular if ϕ has a nonzero ψ T -Fourier coefficient with a non-singular T. ϕ is called singular if ϕ has the property that if a nonzero ψ T -Fourier coefficient F ψ T (ϕ) is nonzero, then det(t ) = 0. Howe shows in 1981 that if an automorphic form ϕ on Sp 2n (A) is singular, then ϕ can be expressed as a linear combination of certain theta functions. Jianshu Li shows in 1989 that any cuspidal automorphic form of Sp 2n (A) is non-singular.

23 Singular Partitions and Rank in the Sense of Howe For a split SO m defined by a non-deg. quad. space (V, q) over F of dim m with the Witt index [ m 2 ], let X+ be a maximal totally isotropic subspace of V with dim [ m 2 ] and let X be the dual to X + : V = X + V 0 + X + with V 0 the orth. complement of X + X + (dim V 0 1). The generalized flag {0} X + V defines a maximal parabolic subgroup P X +, whose Levi part M X + is isomorphic to GL [ m 2 ] and whose unipotent radical N X + is abelian if m is even; and is a two-step unipotent subgroup with its center Z X + if m is odd. When m is even, we set Z X + = N X +.

24 Singular Partitions and Rank in the Sense of Howe By the Pontryagin duality, For any T 2 (F [ m 2 ] ), Z X +(F )\Z X +(A) = 2 (F [ m 2 ] ). ψ T (z(x)) := ψ F (tr(t wx)). For an automorphic form ϕ on G(A), the T or ψ T Fourier coefficient is defined by F ψ T (ϕ)(g) := ϕ(z(x)g)ψ 1 T (z(x))dz(x). Z X + (F )\Z X + (A) An automorphic form ϕ on G(A) is called non-singular if ϕ has a non-zero ψ T -Fourier coefficient with T 2 (F [ m 2 ] ) of maximal rank.

25 Singular Partitions and Rank in the Sense of Howe Denote by p ns the partition corresponding to the non-singular Fourier coefficients. For Sp 2n, p ns = [2 n ], which is a special partition for Sp 2n. For SO 2n+1, one has p ns = { [2 2e 1] if n = 2e; [2 2e 1 3 ] if n = 2e + 1. This is not a special partition of SO 2n+1. For SO 2n, one has p ns = { [2 2e ] if n = 2e; [2 2e 1 2 ] if n = 2e + 1. This is a special partition of SO 2n.

26 Singular Partitions and Lower Bound Partitions J.-Liu-Savin show 2015 that for any automorphic representation π, the set p m (π) contains only special partitions. For SO 2n+1, p ns p m (π) for any π A cusp (SO 2n+1 ). Any p p m (π) as π runs in A cusp (SO 2n+1 ) must be bigger than or equal to the following partition { [32 2e ] if n = 2e; p Gn ns p SO 2n+1 ns = [32 2e ] if n = 2e + 1. denotes the G n -expansion of the partition p ns, i.e., the smallest special partition which is bigger than or equal to p ns. Proposition: For a split classical group G n, p Gn is a lower ns bound for partitions in the set p m (π) as π runs in A cusp (G n ).

27 Singular Partitions and Small Cuspidal Representations It is natural to ask whether the lower bound p Gn ns is sharp? For n = 2e even, and F to be totally real, the Ikeda lifting gives π A cusp (Sp 4e ) with the global Arthur parameter (τ, 2e) (1, 1), where τ A cusp (GL 2 ) is self-dual and has the trivial central character. First, for any p p m (π), one has p [2 2e ] = p Sp 4e. ns Then, one also has p Sp 4e = [2 2e ] p. ns Proposition: If F is totally real, the Ikeda lifting π of Sp 4e has: p Sp 4e = [2 2e ] p, and hence the non-singular partition ns p Sp 4e = p ns = [2 2e ] is the sharp lower bound for A cusp (Sp 4e ). ns Note that the construction of Ikeda lifting does not work when F is not totally real or n = 2e + 1 is odd. What happens in such situations?

28 Singular Partitions and Small Cuspidal Representations J.-Liu (2015): If F is totally imaginary and n 5, the global Arthur packet Π (τ,n) (ɛ,1) (Sp 2n ), with τ A cusp (GL 2 ) self-dual, has no cuspidal members, where ɛ = 1 if n = 2e; and ɛ = ω τ if n = 2e + 1. In this case, the construction of the Ikeda lifting is impossible! Theorem (J.-Liu 2015): 1 For any π A cusp (Sp 2n ), p m (π) = [2 n ] if and only if π is hypercuspidal in the sense of I. Piatetski-Shapiro. 2 For ψ = (τ, 2e) (1, 1) Ψ 2 (Sp 4e ) with τ A cusp (GL 2 ), any cuspidal π Π ψ (Sp 4e ) has that p m (π) = [2 2e ]. 3 For ψ = (τ, 2e + 1) (ω τ, 1) Ψ 2 (Sp 4e+2 ) with τ A cusp (GL 2 ), any cuspidal π Π ψ (Sp 4e+2 ) has that p m (π) = [2 2e+1 ]. 4 For ψ = (τ, 2e + 1) Ψ 2 (Sp 6e+2 ) with τ A cusp (GL 3 ), any cuspidal π Π ψ (Sp 6e + 2) has that p m (π) = [2 3e+1 ].

29 Singular Partitions and Small Cuspidal Representations Theorem (J.-Liu 2015): Assume that F is totally imaginary, and Part (1)of the Conjecture holds. For ψ = r i=1(τ i, b i ) Ψ 2 (Sp 2n ) with τ i A cusp (GL ai ) for i = 1, 2,, r, and 2n = ( r a i b i ) 1, i=1 then there exists a constant N 0 depending on (a 1,, a r ) and (b 1,, b r ), such that if 2n > N 0, the global Arthur packet Π ψ (Sp 2n ) contains no cuspidal member.

30 Singular Partitions and Small Cuspidal Representations Example: Take ψ = (1, b 1 ) (τ, b 2 ) with b 1 1 odd, and with τ A cusp (GL 2 ) of symplectic type and b 2 even. Then N 0 = 8 and the global Arthur packet Π ψ (Sp ) contains no cuspidal members except that (b 1, b 2 ) = (1, 2), (1, 4), (3, 2), or (5, 2). Work in Progress: Characterize and construct small cuspidal representations in general. THANKS!